Dual Techniques for Scheduling on a Machine with Varying Speed

Megow, Nicole; Verschae, José

doi:10.1137/16m105589x

Cited by 19 publications

(47 citation statements)

References 37 publications

(62 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further, the area of scheduling with generalized non-decreasing (completion-) time dependent cost functions, such as minimizing j w j f (C j ), e.g. Epstein et al (2012), Megow and Verschae (2018), Höhn and Jacobs (2015), or even more general job-individual cost functions j f j (C j ), e.g. Bansal and Pruhs (2014), Höhn et al (2018), Cheung and Shmoys (2011), Cheung et al (2017) has received quite some attention.…”

Section: Related Workmentioning

confidence: 99%

Optimal algorithms for scheduling under time-of-use tariffs

et al. 2021

Self Cite

View full text Add to dashboard Cite

We consider a natural generalization of classical scheduling problems to a setting in which using a time unit for processing a job causes some time-dependent cost, the time-of-use tariff, which must be paid in addition to the standard scheduling cost. We focus on preemptive single-machine scheduling and two classical scheduling cost functions, the sum of (weighted) completion times and the maximum completion time, that is, the makespan. While these problems are easy to solve in the classical scheduling setting, they are considerably more complex when time-of-use tariffs must be considered. We contribute optimal polynomial-time algorithms and best possible approximation algorithms. For the problem of minimizing the total (weighted) completion time on a single machine, we present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time slots to be used for preemptively scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for. For preemptive scheduling to minimize the makespan, we show that there is a comparably simple optimal algorithm with polynomial running time. This is true even in a certain generalized model with unrelated machines.

show abstract

Section: Related Workmentioning

confidence: 99%

Optimal algorithms for scheduling under time-of-use tariffs

et al. 2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…This is in contrast to the fact that the classic knapsack problem and its covering version admit pseudo-polynomial time algorithms. Megow and Verschae [17] gave a PTAS for the covering version of KPSC.…”

Section: Related Studiesmentioning

confidence: 99%

Submodular Maximization with Uncertain Knapsack Capacity

Kawase

Sumita

Fukunaga

2019

SIAM J. Discrete Math.

View full text Add to dashboard Cite

We consider the maximization problem of monotone submodular functions under an uncertain knapsack constraint. Specifically, the problem is discussed in the situation that the knapsack capacity is not given explicitly and can be accessed only through an oracle that answers whether or not the current solution is feasible when an item is added to the solution. Assuming that cancellation of the last item is allowed when it overflows the knapsack capacity, we discuss the robustness ratios of adaptive policies for this problem, which are the worst case ratios of the objective values achieved by the output solutions to the optimal objective values. We present a randomized policy of robustness ratio (1 − 1/e)/2, and a deterministic policy of robustness ratio 2(1 − 1/e)/21. We also consider a universal policy that chooses items following a precomputed sequence. We present a randomized universal policy of robustness ratio (1 − 1/ 4 √ e)/2. When the cancellation is not allowed, no randomized adaptive policy achieves a constant robustness ratio. Because of this hardness, we assume that a probability distribution of the knapsack capacity is given, and consider computing a sequence of items that maximizes the expected objective value. We present a polynomial-time randomized algorithm of approximation ratio (1 − 1/ 4 √ e)/4 − ǫ for any small constant ǫ > 0.

show abstract

“…We refer the reader to a number of excellent surveys [3,12] and to the references therein for a more comprehensive literature review. That said, a particularly relevant work is that of Megow and Verschae [14], who developed a PTAS for the generalized objective of minimizing n j=1 v j f (C j ), where f is an arbitrary non-decreasing cost function. Later on, Höhn [12] observed that any instance of the airplane refueling problem can be rephrased in terms of this minimization problem by replacing the term 1/C j in the objective function with 1 − 1/C j .…”

Section: Related Workmentioning

confidence: 99%

A polynomial-time approximation scheme for the airplane refueling problem

Gamzu¹,

Segev

2018

J Sched

View full text Add to dashboard Cite

We study the airplane refueling problem which was introduced by the physicists Gamow and Stern in their classical book Puzzle-Math (1958). Sticking to the original story behind this problem, suppose we have to deliver a bomb in some distant point of the globe, the distance being much greater than the range of any individual airplane at our disposal. Therefore, the only feasible option to carry out this mission is to better utilize our fleet via mid-air refueling. Starting with several airplanes that can refuel one another, and gradually drop out of the flight until the single plane carrying the bomb reaches the target, how would you plan the refueling policy?The main contribution of Gamow and Stern was to provide a complete characterization of the optimal refueling policy for the special case of identical airplanes. In spite of their elegant and easy-to-analyze solution, the computational complexity of the general airplane refueling problem, with arbitrary tank volumes and consumption rates, has remained widely open ever since, as recently pointed out by Woeginger (Open Problems in Scheduling, Dagstuhl 2010, page 24). To our knowledge, other than a logarithmic approximation, which can be attributed to folklore, it is not entirely obvious even if constant-factor performance guarantees are within reach.In this paper, we propose a polynomial-time approximation scheme for the airplane refueling problem in its utmost generality. Our approach builds on a novel combination of ideas related to parametric pruning, efficient guessing tricks, reductions to well-structured instances of generalized assignment, and additional insight into how LP-rounding algorithms in this context actually work. We complement this result by presenting a fast and easy-to-implement algorithm that approximates the optimal refueling policy to within a constant factor.

show abstract

Dual Techniques for Scheduling on a Machine with Varying Speed

Cited by 19 publications

References 37 publications

Optimal algorithms for scheduling under time-of-use tariffs

Optimal algorithms for scheduling under time-of-use tariffs

Submodular Maximization with Uncertain Knapsack Capacity

A polynomial-time approximation scheme for the airplane refueling problem

Contact Info

Product

Resources

About